Cropping two perpendicular curves at their intersections
You can achieve this result by using the Boolean modifier.

Select your large curved mesh, and add a Boolean modifier set to Difference.



This will make the curved wall follow the contours of the arch in the direction of the Normals. You may need to re-calculate them (Ctrl+N) to get the desired result.



Now if you apply the modifier, you can remove the arch mesh and you should end up with a completed single mesh with the desired arch shape in it.